Nonexistence results for a class of fractional elliptic boundary value problems

Details Nonexistence results for a class of fractional elliptic boundary value problems Nonexistence results for a class of fractional elliptic boundary value problems

2012-01-19

arxiv.org/abs/1201.4007v1

Mathematics

Analysis of PDEs

25 pages

Scientific paper

In this paper we study a class of fractional elliptic problems of the form $$ \Ds u= f(x,u) \quad \textrm{in} \O u=0\quad \textrm{in} \R^N \setminus \O,$$ where $s\in(0,1)$. We prove nonexistence of positive solutions when $\O$ is star-shaped and $f$ is supercritical. We also derive a nonexistence result for subcritical $f$ in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the Caffarelli-Silvestre extension \cite{CSilv} of a solution of the above problem. The standard approach in the case $s=1$ using Pohozaev type identities does not carry over to the case $0

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